Section 3 4 Continuity and One sided Limits Answers

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Problem 1

Which of the following statements about the function $y=f(x)$ graphed here are true, and which are false?
a. $\lim _{x \rightarrow-1^{+}} f(x)=1$
b. $\lim _{x \rightarrow 0^{-}} f(x)=0$
c. $\lim _{x \rightarrow 0^{-}} f(x)=1$
d. $\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)$
e. $\lim _{x \rightarrow 0} f(x)$ exists.
f. $\lim _{x \rightarrow 0} f(x)=0$
g. $\lim _{x \rightarrow 0} f(x)=1$
h. $\lim _{x \rightarrow 1} f(x)=1$
i. $\lim _{x \rightarrow 1} f(x)=0$
j. $\lim _{x \rightarrow 2} f(x)=2$
k. $\lim _{x \rightarrow-1^{-}} f(x)$ does not exist.
1. $\lim _{x \rightarrow 2^{+}} f(x)=0$

Carson Merrill

Numerade Educator

Problem 2

Which of the following statements about the function $y=f(x)$ graphed here are true, and which are false?
a. $\lim _{x \rightarrow-1^{+}} f(x)=1$
b. $\lim _{x \rightarrow 2} f(x)$ does not exist.
c. $\lim _{x \rightarrow 2} f(x)=2$
d. $\lim _{x \rightarrow 1^{-}} f(x)=2$
e. $\lim _{x \rightarrow 1^{+}} f(x)=1$
f. $\lim _{x \rightarrow 1} f(x)$ does not exist.
g. $\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{-}} f(x)$
h. $\lim _{x \rightarrow c} f(x)$ exists at every $c$ in the open interval (-1,1) .
i. $\lim _{x \rightarrow c} f(x)$ exists at every $c$ in the open interval (1,3) .
j. $\lim _{x \rightarrow-1^{-}} f(x)=0$
k. $\lim _{x \rightarrow 3^{+}} f(x)$ does not exist.

Matt Just

Numerade Educator

Problem 3

Let $f(x)=\left\{\begin{array}{ll}3-x, & x<2 \\ \frac{x}{2}+1, & x>2\end{array}\right.$
a. Find $\lim _{x \rightarrow 2^{+}} f(x)$ and $\lim _{x \rightarrow 2^{-}} f(x)$.
b. Does $\lim _{x \rightarrow 2} f(x)$ exist? If so, what is it? If not, why not?
c. Find $\lim _{x \rightarrow 4}-f(x)$ and $\lim _{x \rightarrow 4^{+}} f(x)$.
d. Does $\lim _{x \rightarrow 4} f(x)$ exist? If so, what is it? If not, why not?

Stephen Hobbs

Numerade Educator

Problem 4

Let $f(x)=\left\{\begin{array}{ll}3-x, & x<2 \\ 2, & x=2 \\ \frac{x}{2}, & x>2\end{array}\right.$
a. Find $\lim _{x \rightarrow 2^{+}} f(x), \lim _{x \rightarrow 2} f(x)$, and $f(2)$.
b. Does $\lim _{x \rightarrow 2} f(x)$ exist? If so, what is it? If not, why not?
c. Find $\lim _{x \rightarrow-1^{-}} f(x)$ and $\lim _{x \rightarrow-1^{+}} f(x)$.
d. Does $\lim _{x \rightarrow-1} f(x)$ exist? If so, what is it? If not, why not?

Stephen Hobbs

Numerade Educator

Problem 5

Let $f(x)=\left\{\begin{array}{ll}0, & x \leq 0 \\ \sin \frac{1}{x}, & x>0\end{array}\right.$
a. Does $\lim _{x \rightarrow 0^{+}} f(x)$ exist? If so, what is it? If not, why not?
b. Does $\lim _{x \rightarrow 0} f(x)$ exist? If so, what is it? If not, why not?
c. Does $\lim _{x \rightarrow 0} f(x)$ exist? If so, what is it? If not, why not?

Darshan Maheshwari

Numerade Educator

Problem 6

Let $g(x)=\sqrt{x} \sin (1 / x)$
a. Does $\lim _{x \rightarrow 0^{+}} g(x)$ exist? If so, what is it? If not, why not?
b. Does $\lim _{x \rightarrow 0^{-}} g(x)$ exist? If so, what is it? If not, why not?
c. Does $\lim _{x \rightarrow 0} g(x)$ exist? If so, what is it? If not, why not?

Darshan Maheshwari

Numerade Educator

Problem 7

a. Graph $f(x)=\left\{\begin{array}{ll}x^{3}, & x \neq 1 \\ 0, & x=1 .\end{array}\right.$
b. Find $\lim _{x \rightarrow 1^{-}} f(x)$ and $\lim _{x \rightarrow 1^{+}} f(x)$.
c. Does $\lim _{x \rightarrow 1} f(x)$ exist? If so, what is it? If not, why not?

Amy Jiang

Numerade Educator

Problem 8

a. Graph $f(x)=\left\{\begin{array}{ll}1-x^{2}, & x \neq 1 \\ 2, & x=1 .\end{array}\right.$
b. Find $\lim _{x \rightarrow 1^{+}} f(x)$ and $\lim _{x \rightarrow 1^{-}} f(x)$.
c. Does $\lim _{x \rightarrow 1} f(x)$ exist? If so, what is it? If not, why not?

Lucas Finney

Numerade Educator

Problem 9

Graph the functions in Exercises 9 and 10 . Then answer these questions.
a. What are the domain and range of $f ?$
b. At what points $c$, if any, does $\lim _{x \rightarrow c} f(x)$ exist?
c. At what points does only the left-hand limit exist?
d. At what points does only the right-hand limit exist?
$$
f(x)=\left\{\begin{array}{ll}
\sqrt{1-x^{2}}, & 0 \leq x<1 \\
1, & 1 \leq x<2 \\
2, & x=2
\end{array}\right.
$$

Darshan Maheshwari

Numerade Educator

Problem 10

Graph the functions in Exercises 9 and 10 . Then answer these questions.
a. What are the domain and range of $f ?$
b. At what points $c$, if any, does $\lim _{x \rightarrow c} f(x)$ exist?
c. At what points does only the left-hand limit exist?
d. At what points does only the right-hand limit exist?
$$
f(x)=\left\{\begin{array}{ll}
x, & -1 \leq x<0, \text { or } 0<x \leq 1 \\
1, & x=0 \\
0, & x<-1 \quad \text { or } \quad x>1
\end{array}\right.
$$

Darshan Maheshwari

Numerade Educator

Problem 11

Find the limits in Exercises $11-18$.
$$
\lim _{x \rightarrow-0.5^{-}} \sqrt{\frac{x+2}{x+1}}
$$

Stephen Hobbs

Numerade Educator

Problem 12

Find the limits in Exercises $11-18$.
$$
\lim _{x \rightarrow 1^{+}} \sqrt{\frac{x-1}{x+2}}
$$

Darshan Maheshwari

Numerade Educator

Problem 13

Find the limits in Exercises $11-18$.
$$
\lim _{x \rightarrow-2^{+}}\left(\frac{x}{x+1}\right)\left(\frac{2 x+5}{x^{2}+x}\right)
$$

Darshan Maheshwari

Numerade Educator

Problem 14

Find the limits in Exercises $11-18$.
$$
\lim _{x \rightarrow 1^{-}}\left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right)
$$

Darshan Maheshwari

Numerade Educator

Problem 15

Find the limits in Exercises $11-18$.
$$
\lim _{h \rightarrow 0^{+}} \frac{\sqrt{h^{2}+4 h+5}-\sqrt{5}}{h}
$$

Darshan Maheshwari

Numerade Educator

Problem 16

Find the limits in Exercises $11-18$.
$$
\lim _{h \rightarrow 0^{0}} \frac{\sqrt{6}-\sqrt{5 h^{2}+11 h+6}}{h}
$$

Carson Merrill

Numerade Educator

Problem 17

Find the limits in Exercises $11-18$.
a. $\lim _{x \rightarrow-2^{+}}(x+3) \frac{|x+2|}{x+2}$
b. $\lim _{x \rightarrow-2}(x+3) \frac{|x+2|}{x+2}$

Carson Merrill

Numerade Educator

Problem 18

Find the limits in Exercises $11-18$.
a. $\lim _{x \rightarrow 1^{+}} \frac{\sqrt{2 x}(x-1)}{|x-1|}$
b. $\lim _{x \rightarrow 1^{-}} \frac{\sqrt{2 x}(x-1)}{|x-1|}$

Carson Merrill

Numerade Educator

Problem 19

Use the graph of the greatest integer function $y=\lfloor x\rfloor$, Figure 1.10 in Section $1.1,$ to help you find the limits in Exercises 19 and 20 .
a. $\lim _{\theta \rightarrow 3^{+}} \frac{\lfloor\theta\rfloor}{\theta}$
b. $\lim _{\theta \rightarrow 3^{-}} \frac{\lfloor\theta\rfloor}{\theta}$

Stephen Hobbs

Numerade Educator

Problem 20

Use the graph of the greatest integer function $y=\lfloor x\rfloor$, Figure 1.10 in Section $1.1,$ to help you find the limits in Exercises 19 and 20 .
a. $\lim _{t \rightarrow 4^{+}}(t-\lfloor t\rfloor)$
b. $\lim _{t \rightarrow 4^{-}}(t-\lfloor t\rfloor)$

Carson Merrill

Numerade Educator

Problem 21

Find the limits in Exercises $21-42$.
$$
\lim _{\theta \rightarrow 0} \frac{\sin \sqrt{2} \theta}{\sqrt{2} \theta}
$$

Carson Merrill

Numerade Educator

Problem 22

Find the limits in Exercises $21-42$.
$$
\lim _{t \rightarrow 0} \frac{\sin k t}{t} \quad(k \text { constant })
$$

Carson Merrill

Numerade Educator

Problem 23

Find the limits in Exercises $21-42$.
$$
\lim _{y \rightarrow 0} \frac{\sin 3 y}{4 y}
$$

Carson Merrill

Numerade Educator

Problem 24

Find the limits in Exercises $21-42$.
$$
\lim _{h \rightarrow 0^{-}} \frac{h}{\sin 3 h}
$$

Carson Merrill

Numerade Educator

Problem 25

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} \frac{\tan 2 x}{x}
$$

Carson Merrill

Numerade Educator

Problem 26

Find the limits in Exercises $21-42$.
$$
\lim _{t \rightarrow 0} \frac{2 t}{\tan t}
$$

Carson Merrill

Numerade Educator

Problem 27

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} \frac{x \csc 2 x}{\cos 5 x}
$$

Carson Merrill

Numerade Educator

Problem 28

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} 6 x^{2}(\cot x)(\csc 2 x)
$$

Carson Merrill

Numerade Educator

Problem 29

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} \frac{x+x \cos x}{\sin x \cos x}
$$

Carson Merrill

Numerade Educator

Problem 30

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} \frac{x^{2}-x+\sin x}{2 x}
$$

Carson Merrill

Numerade Educator

Problem 31

Find the limits in Exercises $21-42$.
$$
\lim _{\theta \rightarrow 0} \frac{1-\cos \theta}{\sin 2 \theta}
$$

Carson Merrill

Numerade Educator

Problem 32

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} \frac{x-x \cos x}{\sin ^{2} 3 x}
$$

Carson Merrill

Numerade Educator

Problem 33

Find the limits in Exercises $21-42$.
$$
\lim _{t \rightarrow 0} \frac{\sin (1-\cos t)}{1-\cos t}
$$

Carson Merrill

Numerade Educator

Problem 34

Find the limits in Exercises $21-42$.
$$
\lim _{h \rightarrow 0} \frac{\sin (\sin h)}{\sin h}
$$

Carson Merrill

Numerade Educator

Problem 35

Find the limits in Exercises $21-42$.
$$
\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\sin 2 \theta}
$$

Stephen Hobbs

Numerade Educator

Problem 36

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} \frac{\sin 5 x}{\sin 4 x}
$$

Carson Merrill

Numerade Educator

Problem 37

Find the limits in Exercises $21-42$.
$$
\lim _{\theta \rightarrow 0} \theta \cos \theta
$$

Carson Merrill

Numerade Educator

Problem 38

Find the limits in Exercises $21-42$.
$$
\lim _{\theta \rightarrow 0} \sin \theta \cot 2 \theta
$$

Carson Merrill

Numerade Educator

Problem 39

Find the limits in Exercises $21-42$.
$$
\lim _{x \rightarrow 0} \frac{\tan 3 x}{\sin 8 x}
$$

Carson Merrill

Numerade Educator

Problem 40

Find the limits in Exercises $21-42$.
$$
\lim _{y \rightarrow 0} \frac{\sin 3 y \cot 5 y}{y \cot 4 y}
$$

Carson Merrill

Numerade Educator

Problem 41

Find the limits in Exercises $21-42$.
$$
\lim _{y \rightarrow 0} \frac{\sin 3 y \cot 5 y}{y \cot 4 y}
$$

Carson Merrill

Numerade Educator

Problem 42

Find the limits in Exercises $21-42$.
$$
\lim _{\theta \rightarrow 0} \frac{\theta \cot 4 \theta}{\sin ^{2} \theta \cot ^{2} 2 \theta}
$$

Carson Merrill

Numerade Educator

Problem 43

Once you know $\lim _{x \rightarrow a^{+}} f(x)$ and $\lim _{x \rightarrow a^{-}} f(x)$ at an interior point
of the domain of $f$, do you then know $\lim _{x \rightarrow a} f(x) ?$ Give reasons for your answer.

Issa Dababneh

Numerade Educator

Problem 44

If you know that $\lim _{x \rightarrow c} f(x)$ exists, can you find its value by calculating $\lim _{x \rightarrow c^{+}} f(x) ?$ Give reasons for your answer.

Matt Just

Numerade Educator

Problem 45

Suppose that $f$ is an odd function of $x$. Does knowing that $\lim _{x \rightarrow 0^{+}} f(x)=3$ tell you anything about $\lim _{x \rightarrow 0^{-}} f(x) ?$ Give rea-
sons for your answer.

Puneet Prajapati

Numerade Educator

Problem 46

Suppose that $f$ is an even function of $x .$ Does knowing that $\lim _{x \rightarrow 2^{-}} f(x)=7$ tell you anything about either $\lim _{x \rightarrow-2} f(x)$ or $\lim _{x \rightarrow-2^{+}} f(x) ?$ Give reasons for your answer.

Mutahar Mehkri

Numerade Educator

Problem 47

Given $\epsilon>0,$ find an interval $I=(5,5+\delta), \delta>0,$ such that if $x$ lies in $I$, then $\sqrt{x-5}<\epsilon$. What limit is being verified and what is its value?

Regina Hays

Numerade Educator

Problem 48

Given $\epsilon>0,$ find an interval $I=(4-\delta, 4), \delta>0,$ such that if $x$ lies in $I$, then $\sqrt{4-x}<\epsilon$. What limit is being verified and what is its value?

Regina Hays

Numerade Educator

Problem 49

Use the definitions of right-hand and left-hand limits to prove the limit statements in Exercises 49 and 50 .
$$
\lim _{x \rightarrow 0^{-}} \frac{x}{|x|}=-1
$$

Carson Merrill

Numerade Educator

Problem 50

Use the definitions of right-hand and left-hand limits to prove the limit statements in Exercises 49 and 50 .
$$
\lim _{x \rightarrow 2^{+}} \frac{x-2}{|x-2|}=1
$$

Carson Merrill

Numerade Educator

Problem 51

Greatest integer function Find (a) $\left.\lim _{x \rightarrow 400^{+}} \mid x\right\rfloor$ and $(\mathbf{b})$ $\lim _{x \rightarrow 400}\lfloor x\rfloor$; then use limit definitions to verify your findings.
(c) Based on your conclusions in parts
(a) and (b), can you say anything about $\lim _{x \rightarrow 400}\lfloor x\rfloor ?$ Give reasons for your answer.

Carson Merrill

Numerade Educator

Problem 52

One-sided limits Let $f(x)=\left\{\begin{array}{ll}x^{2} \sin (1 / x), & x<0 \\ \sqrt{x}, & x>0\end{array}\right.$
Find (a) $\lim _{x \rightarrow 0^{+}} f(x)$ and (b) $\lim _{x \rightarrow 0^{-}} f(x) ;$ then use limit defini-
tions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about $\lim _{x \rightarrow 0} f(x)$ ? Give reasons for your answer.

Carson Merrill

Numerade Educator

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